where $F(\omega^\prime)$ and $G(\omega^\prime)$ are the Fourier transforms of $f(x)$ and $g(x)$ respectively.

Although I understand the derivation of this formula, I've got difficulty making sense of two frequency terms $\omega$ and $\omega^\prime$. I'm fine with $\omega^\prime$ but I don't know what to make of $\omega$. Should I treat it as a constant, or should I set it to zero?

I'm really interested in the Fourier transform of the square of the second derivative of a function e.g. $\mathcal{F}[ f^{\prime\prime}(x)^2 ]$. Because this problem does not involve a shift, I don't know what to make of the shift term $\omega$.

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1 Answer
1

The notation is not quite right, let F(w) and not F(w') be the Fourier transform of f(x) (resp G(w) for g(x)), so that w is the "frequency term", and w' (or better $\lambda$ to avoid confusion) be just a dummy variable for integration.

Then the convolution integral $\int^{\infty}_{-\infty}{F(\lambda)G(w-\lambda)d\lambda}$ see $\lambda$ as the variable of integration and after integration $\lambda$ will vanish and the result will be a function of w.

For the second question: If $F(w)$ is the Fourier transform of f(x), recall $\frac{d^nf(x)}{dx^n}$ ~ $(2 \pi iw)^n F(w)$.